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American Mathematical Society, Proceedings of the American Mathematical Society, 1(138), p. 37-46
DOI: 10.1090/s0002-9939-09-10067-9
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Congruences for the second-order Catalan numbers
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Abstract
Let p be any odd prime. We mainly show that $∑_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$ and $∑_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.